Monroe Eskew <meskew at math.uci.edu> wrote:
> Look at the axioms of PA. I need not list them here. They are such
> basic and bone-headed statements. They follow from the concept of
> natural numbers. If you disagree with this statement please tell me
> which of the axioms could possibly be false while its negation could
> still purport to describe something deserving of the title "natural
> numbers" (whether fictional or real).
I already said that I don't think a contradiction in PA would "falsify
platonism" (about the integers), but let me say a bit more. In particular
let me try to argue the opposite side a little more forcefully, and then
try to answer it.
As Monroe Eskew says, it's all very well to argue abstractly that a
contradiction in PA would not cause any kind of crisis for platonism, but
if we try to think harder about just what such a contradiction would look
like, I think we can't get off the hook quite as easily as it might seem
at first glance.
First of all, let's be a little more careful than usual about what we mean
by "PA." When I see that abbreviation, I think of first-order arithmetic,
with the induction schema assumed to hold for all first-order formulas.
It's worth reminding ourselves, however, that when a mathematician who has
not specifically studied logic hears the term "Peano arithmetic," he or
she has a less precise object in mind. In particular, the induction axiom
is supposed to hold for "any property" of the natural numbers. We could
refer to this version of the Peano axioms as "second-order PA," but I am
going to resist that term, because the "mathematician in the street"
doesn't know what "second-order" means. I'll use "PA" for the usual
first-order axiom schema and "the Peano axioms" for the vaguer concept.
If you press the mathematician in the street to say exactly what "the
natural numbers" are, it's quite likely that the response will be
something along the lines of, "Any structure that satisfies the Peano
axioms." In particular, the Peano axioms are thought of as *defining* or
*specifying* what the natural numbers are. Unless the mathematician has
been contaminated by some study of logic or philosophy, the natural
numbers will not be regarded as being some ethereal object that we somehow
grasp directly, and some of whose properties we try to capture with the
Peano axioms.
As evidence for what I just said, let's think about what happens when said
mathematician is exposed to PA, and nonstandard models of PA, for the
first time. Typically, the mathematician is confused when it is
demonstrated that there are objects that are *not* isomorphic to the
natural numbers yet still satisfy PA. I claim that the best explanation
for why people get confused about this point is that
1. their initial impression is that PA is just an attempt to write down
the Peano axioms more formally than we usually do; and
2. since they think of the natural numbers as being defined as "anything
satisfying the Peano axioms," it then doesn't make sense for there to
be such things as "nonstandard models."
Now what does all this have to do with platonism? Well, I would argue
that the typical platonist thinks of the natural numbers in the way that I
have just said that "mathematician in the street" does. The natural
numbers are a definite thing and they are specified by the Peano axioms.
Now the Peano axioms talk about *arbitrary properties* of the natural
numbers, and maybe we can't quite formally articulate what that means, but
*surely* the first-order axioms of PA are a special case. Surely?
Well, if we're sure about that, then the inconsistency of PA would indeed
be a bombshell for the platonist. From a subset of the facts that the
natural numbers must, by definition, satisfy, we would have found a
contradiction. The usual conception of the natural numbers must therefore
be incoherent.
As a sociological fact, I think that the inconsistency of PA would cause
consternation among a fair number of mathematicians. I think it would
drive some people who don't normally worry about these things to start
worrying, and probably to be attracted to formalism (in the sense that,
say, Ed Nelson uses that term).
All right, having said all that, why don't I think that an inconsistency
in PA would "falsify platonism"? Well, the way I've laid things out here,
it's easy to see where the loopholes are. The one that stands out to me
is the assumption that all first-order sentences of arithmetic coherently
express legitimate properties of the natural numbers. This assumption
sure seems obvious, but if we had an inconsistency in PA staring us in the
fact, then I think it would seem less obvious. Supposing that the
inconsistency could be avoided by dropping down to a weaker induction
axiom, that would be a tempting route to take. Then we could still claim
that the Peano axioms define the natural numbers, and that induction
applies to "all properties," but that some of the formulas of first-order
arithmetic do not, despite appearances, legitimately express actual
"properties" of the natural numbers. This route is uncomfortable,
perhaps, but mathematicians could probably learn to live with it. After
all, this is approximately the same approach that mathematicians have
taken with the set-theoretic antinomies.
Another obvious loophole is to drop the assumption that the natural
numbers are *defined* by the Peano axioms. This is easy to say, and
perhaps especially easy for a philosopher to say, but I think Monroe Eskew
rightly points out that it is easier said than done. If the Peano axioms
*don't* define the natural numbers, then just what *are* the natural
numbers anyway? Are we going to turn to set theory to rescue us?
Shudder. But what alternative is there? Philosophers may be able to come
up with some suggestions, but I think it would be hard to come up with
something that would satisfy the mathematician in the street.
Tim